One of the most outstanding mysteries of high energy physics is the lack of understanding of the nature and origin of dark matter (DM), which constitutes nearly 85% of the observed matter density in our Universe [1]}.
Usually, it is assumed that DM couples to the Standard Model (SM) with the corresponding coupling being of the order of the weak strength, for a review see [2]}; however, it should be emphasized that the only interaction between
these two sectors confirmed experimentally is gravity.
In recent years considerable interest has been observed in building DM models where the DM interacts with the SM through gravity only.
Such models offer several rather natural mechanisms which can explain production of DM.
For example, DM production during inflation period of rapid expansion in the early Universe through quantum fluctuations [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, the graviton mediated freeze-in production
of DM [17]}, [18]}, [19]}, [20]}, and production of DM through Planck scale
suppressed portals to SM [21]}, [22]}, [23]}, [24]}.
The observed DM relic abundance can be achieved within these models for a wide range of DM masses from sub eV to the Planck mass.
One aspect common in different production mechanisms of gravitational DM is that such production should occur in the very early Universe at late stages of inflation.
Usually, the relevance of the inflationary and reheating dynamics is ignored with a minimal assumption, such as instantaneous reheating. However, it is well known that the inflaton potential sets the initial conditions for the reheating phase which could last longer with a generic equation of state \(w\)  [25]}, [26]}, [27]} affecting DM production [28]}, [29]}, [30]}, [31]}, [32]}, [33]}, [20]}.
Even in the case of non-instantaneous reheating, it is assumed that the inflaton perturbative decay rate \(\Gamma _\phi \)  is constant, and when the Hubble scale \(H\)  becomes of the order of \(\Gamma _\phi \)  the reheating phase ends.
